Optimal Decompositions of Translations of L 2-Functions

Abstract

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space \(L^2({\mathbb{R}}^n)\). Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space \({\mathcal{H}}\), or equivalent the spectral theory of a unitary representation U of the rank-n lattice \({\mathbb{Z}}^n\) in \({\mathbb{R}}^n\). Starting with a non-zero vector \({\psi}\,{\in}\,{\mathcal{H}}\), we look for relations among the vectors in the cyclic subspace in \({\mathcal{H}}\) generated by ψ. Since these vectors \(\{U(k)\psi|k\,{\in}\,{\mathbb{Z}}^n\}\) involve infinite “linear combinations,” the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name L2-independence. This refers to infinite linear combinations of integral translates of a fixed function with l2-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals.